🔢 Percentage Calculator
Five tools covering every percentage question that actually comes up: a share of a total, a share expressed as a percentage, a change between two numbers, applying a percentage, and working backwards from a discounted price.
1. What is X% of Y?
2. X is what percent of Y?
3. Percentage change from X to Y
4. Increase or decrease X by Y%
5. Reverse a discount or a markup
How to work out any percentage, and the three mistakes everyone makes
A percentage is a fraction with the denominator fixed at 100. That is the whole idea. Per cent, from the Latin per centum, means "out of a hundred", so 37% is just another way of writing 37/100, or 0.37. Everything below follows from that one substitution, and most percentage errors come from forgetting it halfway through a calculation.
In practice, almost every percentage question is one of three. You know the percentage and the total and want the part. You know the part and the total and want the percentage. Or you know two numbers and want the change between them. The calculators above handle all three, plus the two that trip people up: applying a percentage to a number, and running a percentage backwards to recover the original figure.
Finding the part: X% of Y
Turn the percentage into a decimal and multiply. To find 20% of 150, write 20% as 0.20 and multiply: 0.20 × 150 = 30.
There is a shortcut worth knowing because it removes the calculator entirely: x% of y is always the same as y% of x. Finding 18% of 50 looks awkward; finding 50% of 18 is nine, instantly, and the answer is the same. It works because both expressions are xy/100. Once you have seen it you will use it constantly, especially for tips.
Finding the percentage: X is what percent of Y
Divide the part by the whole, then multiply by 100. If 25 students out of 200 chose a module, that is 25 / 200 = 0.125, so 12.5%. The order matters and reversing it is the single most common arithmetic slip: the number you are describing goes on top, the number you are describing it against goes on the bottom.
Finding the change between two numbers
Subtract the old value from the new one, divide by the old value, and multiply by 100:
change % = (new − old) / old × 100
Dividing by the old value, not the new one, is what makes this a change rather than a ratio. A price moving from 80 to 100 has risen by 20 / 80 = 25%. The same price moving back from 100 to 80 has fallen by 20 / 100 = 20%. Same twenty units, two different percentages, because the baseline changed. This is not a quirk; it is the most important thing on this page.
Why a 50% fall needs a 100% rise to recover
That asymmetry has real consequences for anyone looking at an investment, a traffic chart or a sales target. A loss and the gain needed to undo it are never the same number:
| Fall | Value left from 100 | Gain needed to recover |
|---|---|---|
| 10% | 90 | 11.1% |
| 20% | 80 | 25.0% |
| 33% | 67 | 49.3% |
| 50% | 50 | 100.0% |
| 75% | 25 | 300.0% |
| 90% | 10 | 900.0% |
The formula is gain = fall / (1 − fall), with both as decimals. It is also why a 20% discount followed by a further 20% discount is not 40% off: the second discount applies to the already reduced price, so the total is 36%.
Running a percentage backwards
This is where most people reach for the wrong operation. If a jacket costs $68 after a 15% discount, the original price is not $68 plus 15%. The discount was taken off the original, not off $68.
$68 represents 85% of the original, so divide rather than multiply: 68 / 0.85 = $80. Checking it forwards, 15% of $80 is $12, and $80 − $12 = $68. Adding 15% to $68 would have given $78.20, which is wrong by more than two dollars.
The same logic strips tax out of a gross total: at a 21% rate, divide the gross by 1.21 to get the net. Calculator five above does both directions, and the sales tax calculator handles the tax case with the tax amount shown separately.
Percent versus percentage point
These are different units and newspapers confuse them weekly. If an interest rate goes from 4% to 6%, it has risen by two percentage points and by 50 percent. Both statements are true and they describe the same move. Use percentage points when you are comparing two percentages to each other, and percent when you are describing relative change. If a politician says unemployment fell 30%, ask whether they mean from 10% to 7% or from 10% to 9.7%.
Three worked examples
A restaurant bill. The total is $84.50 and you want to leave 18%. Flip it: 18% of 84.50 is the same as 84.50% of 18, but easier still is 10% ($8.45) plus half of that ($4.23) plus a bit under a fifth more. The precise figure is $15.21, giving $99.71. The tip calculator also splits it between diners.
A salary rise. You earn $52,000 and are offered $55,640. The change is (55,640 − 52,000) / 52,000 = 7%. If inflation over the same period was 4.5%, the real rise is roughly 2.5 percentage points, or about $1,300 in today's money.
An exam mark. You scored 63 out of 80. That is 63 / 80 = 0.7875, or 78.75%. To reach 85% overall you would have needed 68 marks, five more.
Frequently Asked Questions (FAQs)
How do I calculate a percentage without a calculator?
Build it from 10%. Move the decimal point one place left to get 10%, halve that for 5%, and halve again for 2.5%. Then add the pieces: 17.5% is 10% + 5% + 2.5%. For awkward percentages, use the swap rule — 4% of 75 is the same as 75% of 4, which is 3.
Why is a 20% discount followed by another 20% not 40% off?
Because the second discount is applied to the reduced price, not the original. $100 becomes $80, then $80 becomes $64. That is 36% off in total, not 40%. Successive percentages multiply: 0.8 × 0.8 = 0.64. The same applies to compounding in the other direction.
Can a percentage be more than 100?
Yes, whenever the part is bigger than the baseline. Revenue growing from $2m to $5m is a 150% increase. What cannot exceed 100% is a share of a fixed whole: no single category can be 120% of a total it belongs to, and if a chart says otherwise the categories are overlapping.
What is the percentage change if the starting value is zero?
It is undefined, because the formula divides by the old value. Going from 0 to 40 is not an infinite percentage increase; it is simply an increase of 40 units. Report the absolute change, or use percentage points if both figures are already percentages.
How do I convert a percentage to a fraction or a decimal?
To a decimal, divide by 100: 37% becomes 0.37. To a fraction, put it over 100 and simplify: 37% is 37/100, which will not reduce, while 40% is 40/100 = 2/5. The fraction calculator does the simplifying and shows the steps.
What is the difference between percentage and percentile?
A percentage is a share of a total. A percentile is a rank within a distribution. Scoring 80% on a test means you got four fifths of the marks; being in the 80th percentile means you did better than 80% of the people who sat it. You can score 55% and still be in the 90th percentile if the paper was hard.
Should I average two percentages by adding them and halving?
Only if both come from the same sized group. A 90% pass rate in a class of 10 and a 50% rate in a class of 90 do not average to 70%; the real figure is 9 + 45 = 54 passes out of 100, so 54%. Averaging percentages without weighting them by their group size is one of the easiest ways to publish a wrong number.